<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon">A</font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> implies for <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">A</font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V3">nowhere_dense</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E3:44_1"/>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:44_1"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E5:44_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/>
<a NAME="E6:44_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">F</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>assume </b><a NAME="E7:44_1"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E8:44_1"/><b>then </b>
<font color="Maroon">F</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:44_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E11:44_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E12:44_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E13:44_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="tops_3.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_3.html#T7" target="_self">Th7</a></i>;<br/>
<b>take </b><font color="Maroon">H</font> = <font color="Maroon">G</font> <a href="subset_1.html#K3">`</a> ;<br/>

<a NAME="E14:44_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T17" target="_self">Th17</a>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<b>hence </b><a NAME="E15:44_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E16:44_1"/>
<font color="Maroon">H</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tops_3.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E17:44_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E18:44_1"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/>
<b>hence </b><a NAME="E19:44_1"/>
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<b>thus </b><a NAME="E20:44_1"/>
<font color="Maroon">H</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_3.html#D3" target="_self">Def3</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>

<a NAME="E21:44_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tops_3.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E22:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>;<br/>
<b>hence </b><a NAME="E23:44_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:44"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E3:44"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> &amp; <font color="Olive">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Olive">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:44_2"/>
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:44_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E3:44_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E4:44_2"/><b>then </b>
<font color="Maroon">G</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/>
<a NAME="E5:44_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<b>assume </b><a NAME="E6:44_2"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E7:44_2"/><b>then </b>
<font color="Maroon">G</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:44_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E10:44_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E11:44_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E12:44_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="tops_3.html#D2" target="_self">Def2</a></i>;<br/>
<b>take </b><font color="Maroon">H</font> = <font color="Maroon">F</font> <a href="subset_1.html#K3">`</a> ;<br/>

<a NAME="E13:44_2"/>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tops_3.html#T17" target="_self">Th17</a>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<b>hence </b><a NAME="E14:44_2"/>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E15:44_2"/>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E16:44_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/>
<b>hence </b><a NAME="E17:44_2"/>
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E18:44_2"/>
<font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_3.html#D3" target="_self">Def3</a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>

<a NAME="E19:44_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="tops_3.html#T26" target="_self">Th26</a>, </i>;<br/>
<b>hence </b><a NAME="E20:44_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


</div><b>end;</b></div>
<a NAME="E4:44"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V3">nowhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E5:44"/>
<font color="Maroon">A</font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 <b>by </b><i/>;<br/>


</div>
